The Cohen-Macaulay Property of $f$-ideals

Abstract

For positive integers $d<n$, let $[n]_d=\{A\in 2^{[n]}\mid |A|=d\}$ where $[n]=:\{1,2,\ldots, n\}$. For a pure $f$-simplicial complex $Δ$ such that ${\rm dim}(Δ)={\rm dim}(Δ^c)$ and $\mathcal{F}(Δ)\cap \mathcal{F}(Δ^c)=\emptyset$, we prove that the facet ideal $I(Δ)$ is Cohen-Macaulay if and only if it has linear resolution. For a $d$-dimensional pure $f$-simplicial complex $Δ$ such that $Δ'=:\langle F\mid F\in [n]_d\smallsetminus \mathcal F(Δ)\rangle$ is an $f$-simplicial complex, we prove that $I(Δ^c)$ is Cohen-Macaulay if and only if $I(Δ')$ has linear resolution.